Centripetal and Centrifugal Forces

The centrifugal effect is the sensation of a force, but is really a result of inertia and the centripetal force.

'Centripetal' comes from Latin. It combines 'centrum' with the verb 'petere', to refer to a force that "seeks the centre". It is the centripetal force that causes an object with constant speed to keep changing direction, so that the object moves in a circular path.

A circular motion involves constant speed but continuously changing direction of the velocity vector. Circular motion therefore has acceleration towards the centre of the curvature.

Centripetal force increases with rotational speed and mass. However, as the radius of curvature increases, the rate of change of velocity decreases, so the centripetal force is inversely proportional to the radius.

'Centrifugal' also comes from Latin, with the verb 'fugere', meaning "to flee the centre". It is what is experienced by the passengers in a car when a car goes around a bend or in a circle. The car pushes against the passengers, whose inertia would otherwise cause them to continue moving in a straight line. The passengers feel a 'force', but it is really the result of the centripetal force causing them to change direction as they go around a circle. Since it is not really a force, we should more properly call it the 'centrifugal effect'.

A car on a curve

When a car goes around a curve, it is accelerating towards the centre of a circle, and the passengers feel a force pushing them towards the outside of the vehicle.

Even though we do not think of it as such, a vehicle going around a bend in the road is actually undergoing circular motion that has a centre of curvature. The car's inertia causes the tyres to push against the road. That is why tyres need to be rough, to grip the road surface and create the friction which pushes the car at right angles to its motion.

Any object or person inside the vehicle will feel the centrifugal force - they 'sense' a push towards the outside of the car. It is not really a force. It is caused by their inertia, which wants to take them in a straight line. It is the vehicle which experiences the force of the road contact, not the passengers. Hence, it is the vehicle which moves under the passengers, not the passengers which move inside the vehicle, creating the illusion of a 'force'.

A car executing a curve moves because of the force between its tyres and the road. An object on the dashboard will continue in a straight line until it hits the outside edge of the car, where it experiences the centripetal force of the car,

The Maths

ac = v2/r, so therefore the force is:
Fc = mv2/r

The centripetal force is equal to the centrifugal force on an object in circular motion. This force, Fc, is equal to the mass times the square of the tangential velocity, v, and inversely proportional to the radius, r.

A Thought Experiment

A thought experiment is one done with the imagination. When it would be too difficult or impossible to do a real experiment, sometimes scientists think through an experiment with a logical argument to demonstrate some principle. Galileo made a famous thought experiment about gravity, which was eventually done by David Scott on the Moon. Albert Einstein proposed many thought experiments to test his Relativity theories, involving trains travelling close to the speed of light and boxes falling through space.

Newton's Cannon

Newton imagined a cannon on a high mountain. There would be a velocity of the cannonball which would cause it to travel fast enough to continually miss the Earth. Even though it is continuously falling towards the ground it can never reach

Isaac Newton also made a thought experiment about gravity and orbits: he imagined a very powerful cannon on top of a very high mountain. Then he asked the question: what would happen if the cannon fired a cannonball parallel to the ground? The answer was it would travel some distance while it fell to the ground. Then he asked what would happen if we make the cannon more and more powerful?

Gradually, the ball would travel further and further, until it was travelling around the curve of the planet's surface, 'missing' more and more of the planet. At a certain point, reasoned Newton, the cannonball would miss the entire planet, and since it still had the original velocity would pass the cannon and go around the planet again... and again. This was the first understanding of 'orbits'.

Orbits

Just like Newton's cannonball, a satellite is continuously falling towards the Earth, but its horizontal velocity is so great it moves around the Earth. The Moon does the same, and the Earth, and all the planets, are in orbit around the Sun. The shape of the orbits is an ellipse. This is like a slightly stretched circle. But the principle of Newton's wonderful thought experiment still applies.

A satellite in orbit is in freefall, and is falling towards the Earth all the time. But is has a velocity that takes it at right angles to the fall. Therefore, it keeps moving to the side at just the right speed that it just misses all the time (it helps that the Earth is round - going around sharp corners would be a bit trickier).

Therefore, any object at the same altitude would fall at the same rate, and therefore have the same period of orbit.

Geo-Synchronous Orbits

The ISS International Space Station orbits the Earth every 92 minutes, at an altitude of about 400-420 km (remember, orbits are elliptical, so their altitude must vary as they go around).

Satellites in geostationary orbit are always over the same point of the Earth's surface (very useful for communications satellites).

Geostationary satellites are synchronised with the rotation of the Earth. Therefore, their orbit is the length of an Earth day, 24 hours.

Since they take a lot longer to go around the Earth than the ISS, their orbit is much higher. In fact, the geosynchronous orbit is at an altitude of 35,564 km (their orbit has a radius of 42,164 km, and the Earth's radius is 6,400 km).

Parabolic Trajectories

Constant horizontal velocity and constant vertical acceleration produce a curve in the shape of a parabola.

Newton's Second Law of Motion says that a projectile will accelerate when in freefall. It also says that a projectile with horizontal velocity would experience exactly the same vertical acceleration as an object in freefall. The result is a parabola.

A student demonstrating that a ball follows a parabolic flight trajectory. The lengths of the strings were calculated from the kinematic equation: $d = 1/2⋅gt^2$, using 0.05s time intervals.